Reduction In Sum Of Squares Research Articles

Exploring Temporal Variability in the Southern Benguela Ecosystem Over the Past Four Decades Using a Time-Dynamic Ecosystem Model

This paper presents a new, updated, temporally-dynamic Ecosim model for the Southern Benguela (1978-2015) by capturing recent advancements in our understanding of the dynamics and newly available data series from an additional decade of research in the region. The paper documents the model development and the thinking behind incorporating a newly available upwelling index, and observed fishing effort as important drivers of modelled ecosystem dynamics. The most sensitive predator-prey interactions were identified and vulnerabilities (of prey to predators) were estimated to improve model fit to field observations under each scenario. Sardine interactions with prey and predators were consistently found to be sensitive interactions in model fitting, accounting for at least 40% of the most sensitive trophic interactions in the southern Benguela modelled food web. Model fits to data were improved by a total of 85% reduction in sum of squares when upwelling effects on large phytoplankton availability to zooplankton and small pelagic fish was incorporated, geographic shifts in sardine distribution were captured by means of altered availability of sardine to predators, corresponding vulnerabilities of prey to predators were estimated, and an additional, small, hypothetical forcing function was fitted to small phytoplankton production. African penguin and Cape gannet fits to data series were improved by incorporating a recently published bird Food Availability Index, although model fits of several fish groups then deteriorated, emphasising the need for additional empirical species-specific functional response studies. Attempts at fitting the full area model to either west coast or south coast time series did not improve overall model fits, highlighting limitations of the current non-spatialized model in describing coast-specific dynamics. This study shows promise in unravelling the observed dynamics of the Benguela upwelling ecosystem, and points to the importance of exploring spatially disaggregated approaches, in particular the new foraging habitat capacity of the Ecospace approach, to improve our understanding of processes whereby variability in upwelling influences dynamics of the Southern Benguela ecosystem. This is important in reconciling knowledge needed to manage fisheries and to protect marine biodiversity by means of ecosystem-based management in South Africa, and to advance management advice under future scenarios of climate change.

Variance components for two-way nested design data

This paper discusses the use of projections for the sums of squares in the analyses of variance for two-way nested design data. The model for this data is assumed to only have random effects. Two different sizes of experimental units are required for a given experimental situation, since nesting is assumed to occur both in the treatment structure and in the design structure. So, variance components are coming from the sources of random effects of treatment factors and error terms in different sizes of experimental units. The model for this type of experimental situation is a random effects model with more than one error terms and therefore estimation of variance components are concerned. A projection method is used for the calculation of sums of squares due to random components. Squared distances of projections instead of using the usual reductions in sums of squares that show how to use projections to estimate the variance components associated with the random components in the assumed model. Expectations of quadratic forms are obtained by the Hartley’s synthesis as a means of calculation.

Analysis of Variance Components for Genetic Markers with Unphased Genotypes.

An ANOVA type general multi-allele (GMA) model was proposed in Wang (2014) on analysis of variance components for quantitative trait loci or genetic markers with phased or unphased genotypes. In this study, by applying the GMA model, we further examine estimation of the genetic variance components for genetic markers with unphased genotypes based on a random sample from a study population. In one locus and two loci cases, we first derive the least square estimates (LSE) of model parameters in fitting the GMA model. Then we construct estimators of the genetic variance components for one marker locus in a Hardy-Weinberg disequilibrium population and two marker loci in an equilibrium population. Meanwhile, we explore the difference between the classical general linear model (GLM) and GMA based approaches in association analysis of genetic markers with quantitative traits. We show that the GMA model can retain the same partition on the genetic variance components as the traditional Fisher's ANOVA model, while the GLM cannot. We clarify that the standard F-statistics based on the partial reductions in sums of squares from GLM for testing the fixed allelic effects could be inadequate for testing the existence of the variance component when allelic interactions are present. We point out that the GMA model can reduce the confounding between the allelic effects and allelic interactions at least for independent alleles. As a result, the GMA model could be more beneficial than GLM for detecting allelic interactions.

Nitrate in Groundwater of the United States, 1991−2003

An assessment of nitrate concentrations in groundwater in the United States indicates that concentrations are highest in shallow, oxic groundwater beneath areas with high N inputs. During 1991-2003, 5101 wells were sampled in 51 study areas throughout the U.S. as part of the U.S. Geological Survey National Water-Quality Assessment (NAWQA) program. The well networks reflect the existing used resource represented by domestic wells in major aquifers (major aquifer studies), and recently recharged groundwater beneath dominant land-surface activities (land-use studies). Nitrate concentrations were highest in shallow groundwater beneath agricultural land use in areas with well-drained soils and oxic geochemical conditions. Nitrate concentrations were lowest in deep groundwater where groundwater is reduced, or where groundwater is older and hence concentrations reflect historically low N application rates. Classification and regression tree analysis was used to identify the relative importance of N inputs, biogeochemical processes, and physical aquifer properties in explaining nitrate concentrations in groundwater. Factors ranked by reduction in sum of squares indicate that dissolved iron concentrations explained most of the variation in groundwater nitrate concentration, followed by manganese, calcium, farm N fertilizer inputs, percent well-drained soils, and dissolved oxygen. Overall, nitrate concentrations in groundwater are most significantly affected by redox conditions, followed by nonpoint-source N inputs. Other water-quality indicators and physical variables had a secondary influence on nitrate concentrations.

Testing for Threshold Autoregression

We consider the problem of determining whether a threshold autoregressive model fits a stationary time series significantly better than an autoregressive model does. A test statistic $\lambda$ which is equivalent to the (conditional) likelihood ratio test statistic when the noise is normally distributed is proposed. Essentially, $\lambda$ is the normalized reduction in sum of squares due to the piecewise linearity of the autoregressive function. It is shown that, under certain regularity conditions, the asymptotic null distribution of $\lambda$ is given by a functional of a central Gaussian process, i.e., with zero mean function. Contiguous alternative hypotheses are then considered. The asymptotic distribution of $\lambda$ under the contiguous alternative is shown to be given by the same functional of a noncentral Gaussian process. These results are then illustrated with a special case of the test, in which case the asymptotic distribution of $\lambda$ is related to a Brownian bridge.

Detection of Heterogeneous Variance in Herd Production Groups

The detection of heterogeneous variance across herd groups is based largely on data stratified by mean herd production. Such stratification is analogous to selection on herd means. This introduces the possibility of biased estimates of genetic and residual variances. Results presented show that Henderson's Method 3 estimates of the residual variance are unbiased by selection on herd means. However, reductions in sums of squares for sires have different expected values in no selection and selection models. The magnitude of this bias is evaluated with a small-scale simulation. Extension of these results to other methods of variance estimation reveals a potential bias from selection on herd means.

A comparison of various methods to scale soil hydraulic properties

Soil water characteristic curves and hydraulic conductivity functions at 57 locations were scaled by various methods. Optimum scaling results were obtained when pressure head and conductivity data were scaled by different methods. Criteria that supported this conclusion were: (1) the percent reduction in sum of squares; (2) the correlation between unscaled and estimated pressure head values as obtained from the scaled mean hydraulic functions; and (3) correlation between scaling factors determined from soil water characteristic data and those determined from hydraulic conductivity data. Both sets of scaling factors were found to be lognormally distributed. A correlation of R 2 = 0.761 was obtained between scaling factors determined from soil water characteristic data and those determined from hydraulic conductivity data. For the optimum scaling methods, the sum of squares, about the average curve, was reduced by 73 and 42%, for the water characteristic and hydraulic conductivity data, respectively. The results suggest that scaling can successfully be used for describing the variability of soil hydraulic properties of different soil map units and horizons.

A Mixed-Effects Model for Categorical Data

A mixed model for categorical data from unbalanced designs which is directly analogous to a two-way ANOVA model for quantitative data is proposed. An extension of the fitting constants method is developed to estimate model variance components based on appropriate reductions in sums of squares. The resulting variance component estimators are incorporated into the covariance structure of a general linear models Wald statistic to test for treatment differences. These procedures are illustrated with data obtained from a multicenter clinical trial in which the treatments are regarded as fixed effects and the clinics are regarded as random effects.

Relationships Among Sires in Estimating Genetic Variance

A method to account for relationships among sires in estimating covariance among paternal half-sibs and heritability is described. The reduction in sum of squares from fitting sire effects after eliminating all other effects in the model is divided by the sum of products of elements of the matrix of coefficients after absorption and elements of the matrix of relationships among sires rather than by the trace of the matrix of coefficients. The method requires computation of relationships that for many sires may be difficult.

Analysis of Variance Techniques for Unbalanced Data

Increasing interest has been expressed, particularly by social scientists, in the analysis of unbalanced data occurring in models not of full rank, i.e., linear models where the numbers of observations in the subclasses are unequal. The various discussions of this problem that have appeared in the psychological and educational literature differ depending upon the intent of the author(s). For example, some writers (Gourlay, 1955; Steinhorst & Miller, 1969; Tsao, 1942, 1946; Williams, 1972) have examined and at times compared different types of solutions, including approximate solutions, for the unbalanced case. Other writers (Overall & Spiegel, 1969; Rawlings, 1972) have focused upon the difficulty which arises in interpreting the results of unbalanced data analyses because the estimable functions involved in tests of hypotheses are not orthogonal. Though not specifically addressed to the unbalanced case but important in this regard is the work of Bottenberg and Ward (1960), Cohen (1968), Hurst (1970), and Jennings (1967) that demonstrates the equivalence of linear regression and the fixed-effects analysis of variance by the use of regression on dummy variables. This diversity in purpose, combined with the relative narrowness of the individual efforts, has resulted in a fragmented treatment of the problem of unbalanced data and in some cases confusion and controversy regarding methodology, e.g., the role of constraints in obtaining a solution for models not of full rank or the calculation of reductions in sums of squares (Overall & Spiegel, 1973; Rawlings, 1973; Smith, 1973). In response, at least one writer (Joe, 1971) has been prompted to call for a greater understanding and more detailed presentation of the models employed. Some of the more important issues encountered in the analysis of unbalanced data will be examined here within a single framework. These issues are broadly defined as (1) solutions to the normal equations for linear models where the incidence or design matrix is not of full column rank, (2) estimable functions of the parameters and tests of hypotheses, and (3) reductions in sums of squares and interpretation of results. Because of the increased availability of computer hardware and sophisticated least squares routines, the various approximate solutions for analyzing unbalanced data are not considered here.

Variances of Variance-Component Estimators for the Unbalanced 2-Way Cross Classification with Application to Balanced Incomplete Block Designs

"Best" estimators of variance components for the unbalanced cases of random-effects models are not known. In fact, even for the very simplest of the unbalanced "designs", the balanced incomplete block designs, the question of the existence of minimum variance unbiased estimators remains open (Kapadia and Weeks [5]). The traditional approach to the derivation of variance-component estimators for unbalanced cases has been to pick several quadratic functions of the data, set these functions equal to their expectations, and then solve the resulting system of equations for the variance components. Two of the estimators derived in this fashion for the variance components associated with the unbalanced two-way cross classification are those referred to as the Methods-1 and -3 estimators of Henderson [4]. Method-1 utilizes quadratics analogous to the sums of squares in a balanced analysis of variance. The quadratics employed in Method-3 represent differences between reductions in sums of squares due to fitting different models. Since in Method-3 more differences between reductions are available than one has variance components to estimate, the method is not uniquely defined. Here, the Method-3 estimators of the components associated with the two-way classification are taken to be those in Harville [3], which are the ones most commonly used. Searle [9] obtained algebraic expressions for the sampling variances of the Method-1 estimators of the "two-way" components. Low [6] gave similar expressions for the Method-3 estimators for the zero-interaction case. Their results were obtained by applying well-known formulas for the variances and covariances of quadratic functions of multivariate-normal random variables. These formulas state that if $\mathbf{y}$ is a random vector having the multivariate normal distribution with mean $\mathbf{u}$ and variance-covariance matrix $\mathbf{V}$ and if $\mathbf{A}$ and $\mathbf{B}$ are square symmetric matrices of appropriate dimension having fixed elements, then \begin{align}\tag{1}\operatorname{var} \lbrack\mathbf{y}'\mathbf{Ay}\rbrack &= 4\mathbf{u}'\mathbf{AV}\mathbf{Au} + 2\mathrm{tr}(\mathbf{VA})^2\end{align}and\begin{align}\tag{2} \operatorname{cov} \lbrack\mathbf{y}'\mathbf{Ay},\mathbf{y}' \mathbf{B}y\rbrack &= 4\mathbf{u}'\mathbf{AVBu} + 2 \mathrm{tr} (\mathbf{VAVB})\end{align} Searle [8], [10] and Mahamunulu [7] have also used these formulas to obtain algebraic expressions for the variances of commonly-used estimators of the components of variance associated with other unbalanced classifications. In the present paper, results (supplementary to those of Searle) are given which lead to expressions for the sampling variances of Method-3 estimators of the variance components associated with the unbalanced two-way cross classification with interaction. By using these results in combination with those of Searle, the variances of Method-1 and Method-3 estimators can be directly compared for a given set of subclass numbers. The results are shown to simplify when the "unbalancedness" is of the type associated with a balanced incomplete block design. Neither estimator of any component is uniformly better than the other for any such design. (Except for the estimators of the residual component which are identically equal.)